Psyc_315_-_Winter_2010_-_Class_11_Review_

Psyc_315_-_Winter_2010_-_Class_11_Review_ - Z-Scores Z...

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1 Z-Scores • Z score is a number that is commonly used to convey the relative position relative position of a data value in the data set. • The z-score score is obtained by subtracting the mean of the data set from the value and dividing the result by the standard deviation of the data set. • The Z score designates how many standard deviations the corresponding raw score is above or below the mean. X M Z SD - = 2 Z-Scores • A Z score tells you two things about a score: 1.Its algebraic sign indicates whether the score is above or below the mean 2.Its absolute value tells you by how much. Z scores provide a helpful way to compare scores that are on completely different scales or from different distributions, because the mean of Z scores are always 0, and the standard deviation 1 . 3 Example 1 What is the z -score for the value of 14 in the following score for the value of 14 in the following sample values? X = [3, 8, 6, 14, 4, 12, 7, 10] Mx Mx = 8 = 8 SDx = 3.57 • Thus, the data value of 14 is 1.68 standard deviations above the mean of 8, since the z-score is positive. 14 8 1.68 3.57 X M Z SD - - = = =
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4 Example 2 • What is the z What is the z -score for the value of 95 in the score for the value of 95 in the following values? following values? X = [96, 114, 100, 97, 101, 102, 99, 95, 90] ( ) 2 X X M X M M SD Z N N SD - - = = = 5 Example 2 Answers X = [ 96 114 100 97 101 102 99 95 90 96 114 100 97 101 102 99 95 90], so the mean of X, M = 99.33 (M = X/ N = 894/9). (X - M) = [-3.33, 14.67, 0.67, -2.33, 1.67, 2.67, -0.33, -4.33, -9.33] • Deviation score, subtract M from each number in X. (X – M) 2 = [11.09, 215.21, 0.45, 5.43, 2.79, 7.13, 0.11, 18.75, 87.05] • Squared deviations from the mean, M. SS= Σ (X - M ) 2 = 348.01 • Sum of squared deviations from the mean (SS). SD 2 = SS / N = 38.67 • Average squared deviation from the mean. SD= SD 2 = 6.22 • Square root of variance. 6 Example 2 Answers • First compute the mean and standard deviation. First compute the mean and standard deviation. These values are respectively 99.33 & 6.22. These values are respectively 99.33 & 6.22. • Z-Score Score = (95 = (95 –99.33)/6.22 99.33)/6.22 = -0.70 0.70 . • Thus, the data value of 95 is located 0.70 Thus, the data value of 95 is located 0.70 standard deviation standard deviation below below the mean value of the mean value of 99.33 (since the z 99.33 (since the z -score is negative). score is negative).
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7 Example 3 M = 23; SD 2 = 1.56. Calculate the raw scores for the following z-scores
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Psyc_315_-_Winter_2010_-_Class_11_Review_ - Z-Scores Z...

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